For mobile devices, GreenBits and Samourai Wallet support SegWit addresses. GreenBits is officially available, but Samourai Wallet isn't officially released yet, but you can still get it to try it out.

AirBitz is in the process of rebranding themselves to "Edge" and will have SegWit address support when that is released.

For desktop use, Electrum 3.0 isn't out yet but will have SegWit address support.

The latest version of the Bitcoin Core client supports generating SegWit addresses on the console, and will support them in the GUI in 0.15.1.

There are different types of wallets. You can choose between online-, hardware-, lightweight-, mobile- and classic (full node) wallets.The transaction fee is not dependend on the wallet you use. Its depending on how much transaction are waiting for confirmations.You can check the current average fee to get your transaction confirmed within x blocks here: https://bitcoinfees.21.co.

What you are basically looking for is a wallet with adjustable fees.If you are looking for a wallet for your desktop pc i would recommend electrum as a lightweight wallet (does not need whole blockchain).You can adjust fee as you wish within electrum.If you want a mobile wallet i would recommend mycelium. The fee is "pretty adjustable". You cant set it manually, but you got a lot of optionsto chose from. Those range from 10 B/Sat to 300+ B/Sat.If you are going to store big amounts of bitcoins you should use a paper wallet or (for better usability) a hardware wallet. A more detailed list of bitcoin wallets can be found here: https://www.weusecoins.com/en/find-the-best-bitcoin-wallet/